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Problem of diffraction at a fine screen. (English. Russian original) Zbl 0556.35117

Sib. Math. J. 25, 31-42 (1984); translation from Sib. Mat. Zh. 25, No. 1(143), 39-52 (1984).
Let \({\mathcal D}\subset {\mathbb{R}}^ 3\) be an open domain with smooth boundary \(\partial {\mathcal D}\) and let \(S\subset {\mathcal D}\), dim S\(=2\) be a smooth surface with boundary \(\Gamma =\partial S\). Consider the problem \[ (1)\quad (\Delta +k^ 2)U=0\quad in\quad {\mathcal D}\setminus S,\quad \lambda U=\pm (1/\sigma)\partial U/\partial n+H\quad on\quad S_{\pm},\quad \partial U/\partial \omega +\beta U=0\quad on\quad \partial {\mathcal D}. \] Here k, \(\lambda\in {\mathbb{C}}\), \(\beta\) is a smooth function, \(\omega\) is a vector field transversal to S and \(S_{\pm}\) are two sides of S. One assumes that U satisfies the so called Meixner condition near \(\Gamma\), that is \(| \nabla U|^ 2\) near \(\Gamma\) is integrable and the energy is finite. Introducing in a neighbourhood T(\(\Gamma)\) of \(\Gamma\) new coordinates \(Z=(z_ 0,z)\), \(z=(z_ 1,z_ 2)\) so that \(S\cap T(\Gamma)=\{z_ 2=0,z_ 1\leq 0\},\) \(\Gamma =\{z=0\}\), the author obtains an expression for the Laplacian near \(\Gamma\). To study the existence and the regularity of the solution, some Sobolev type spaces \(E^ s({\mathcal D}\setminus S)\), \(E^ s(S_+\cup S_-)\) are introduced.
The main result states that the problem (1) has a unique solution \(U\in E^{s+2}({\mathcal D}\setminus S),\) \(s\geq 0\), provided \(H\in E^{s+}(S_+\cup S_-),\) for all \(\lambda\) with exception of a countable set. The proof is based on the analysis of second order elliptic operators with polynomial coefficients by using pseudodifferential operators.
Reviewer: V.Petkov

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
78A45 Diffraction, scattering
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
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